The physical basis of osmosis

It is surprising that osmosis, a phenomenon so central to biology, has been cloaked in misunderstanding for so long. The authors show that the most plausible account for what drives water fluxes is one put forward by Peter Debye in 1923, where the repulsion of solute molecules from the semipermeable membrane generates a pressure drop, which draws water from a chamber with low solute concentration to one that is high.


Introduction
Osmosis is one of the most powerful forces that organisms must counteract to survive. An index of its importance is that animal cells, of all kinds, spend about a quarter of their energy resisting the osmotic challenge induced by the presence of impermeant molecules in cells (i.e., the Donnan effect, Appendix 1; Rolfe and Brown, 1997;Kay, 2017). An unchecked Donnan effect would lead to a continuous influx of water until the cell bursts. The need to maintain osmotic balance is unrelenting, interrupted neither by sleep nor hibernation. Furthermore, osmosis is quite literally at the root of plant physiology (Niklas and Spatz, 2012).
The phenomenological thermodynamics of osmosis has long been clear, at least for osmotic equilibrium. van 't Hoff's equation for the equilibrium pressure difference can be derived by equating the chemical potentials of the water in the two compartments (Dill and Bromberg, 2003;Phillips et al., 2012) separated by a semipermeable membrane, but this thermodynamic derivation provides no insight into the molecular mechanism that generate the pressure difference. Indeed, the molecular basis of osmosis continues to be widely mischaracterized and hence misunderstood, although a consistent mechanistic understanding was presented 100 yr ago (Debye, 1923b).
In this paper, we will show why a molecular basis for osmosis that is most often given in biology textbooks is invalid. This misconception consists in the belief that the osmotic water flux is driven by a gradient in water concentration across the membrane. We will show how the osmotic and diffusive fluxes of water can be separately measured across a semipermeable membrane. This can then be used to demonstrate that diffusion alone cannot account for the osmotic flux across membranes with aqueous pores. We will then show how a physical mechanism that was first presented by Peter J.W. Debye in 1923 can generate a macroscopic pressure and provides the most plausible account of osmosis. We refer to it as the "Debye model." Debye was perhaps the first to recognize that osmosis arises from the mechanical interaction of an impermeant solute with a semipermeable membrane but does not depend on the precise chemical nature of the solute or the solvent. We believe that the Debye model has failed to take hold in biology for several reasons, inter alia, a lack of understanding of the physical argument, its requirement for mathematical explication, and the availability of other simple, seemingly plausible, but flawed arguments. In addition, textbooks, besides omitting the Debye model, have not raised any inconsistencies in the conventional approach. There has hence seemed little need to question what at first blush seems a simple phenomenon.
There have been several attempts primarily directed at biologists to set the record straight on the physical basis of osmosis (Stein, 1966;Kramer and Myers, 2012), as well as accounts of the Debye model in journals (Manning, 1968;Oster and Peskin, 1992;Borg, 2003 Preprint;Marbach and Bocquet, 2019;Song et al., 2021) and textbooks (Villars and Benedek, 1974;Weiss, 1996;Baumgarten and Feher, 2011;Nelson, 2014), but despite these efforts, misconceptions have persisted. The apparent simplicity of osmosis may have masked what is at bottom a rather subtle phenomenon with enormous implications for biology (Dick, 1966;House, 1974;Andersen, 2015). It is, we think, worth readdressing the physical basis of osmosis because it may open new ways of looking at water and solute transport that have remained hidden because of flawed beliefs, and it is important to ensure that our understanding is firmly rooted in wellestablished physical principles.
The osmotic flux of water is important in several biological disciplines; indeed, it is a challenge to find one where it is not. However, different branches of science have developed unique terminologies, which may confuse someone familiar with the terms of one field in reading the literature of another. The unified view and terminology presented here may help to bring consilience to the study of osmosis.
We first provide a review of the basic empirical information about osmosis, including a discussion of some misconceptions. Then, we give an account of the Debye model, both as presented by Debye himself to derive van 't Hoff's law for osmotic equilibrium, and as extended to apply to osmotic flow (Manning, 1968). The seminal contributions of the Norwegian physicist Lars Vegard are integrated into this account (Vegard, 1908). Finally, we discuss water flow across biological membranes in the context of Debye's model.

The rudiments of osmosis and common misconceptions
To illustrate the process of osmosis, we will consider a semipermeable membrane, namely, one that is permeable to water but completely impermeable to solute molecules, separating two solutions. We will restrict our discussion to water, but it also applies to any other solvent. If the osmolarities (i.e., the total concentrations of solutes in moles per unit volume) of the solutions differ, water flows from the solution with the lower osmolarity to that with the higher osmolarity. In the situation diagrammed in Fig. 1, the movement of water can be prevented if the piston exerts excess pressure on the solution with higher osmolarity equal, if the solutions are dilute, to RTΔc s , where Δc s is the osmolarity difference. This experimental observation is encapsulated by van 't Hoff's equation, where ΔP is the pressure difference under no-flow, equilibrium conditions between two solution chambers separated by a semipermeable membrane. 1 The definitions of symbols in the equations can be found in Table 1. The term RTc s in a free-standing solution with solute concentration c s is often referred to as the "osmotic pressure" of the solution. However, this imprecision is the source of some confusion since an actual pressure difference can only arise between two solutions with different osmolarities separated by a semipermeable membrane. It is worth emphasizing that osmotic pressure is not a physical property of a free-standing aqueous solution.
We will lay out our argument in terms of the osmolarities of the solutions. However, to understand the osmotic flux of water in cells, it is important to consider that macromolecules in both the cytoplasm and extracellular solutions may exclude water. The osmotically active solute concentration within a cell is determined by the number of moles of solute per mass of freely exchangeable water molecules, namely, the osmolality (Boron and Boulpaep, 2016). For dilute solutions, which we are considering, the osmolarity and osmolalities are essentially identical.
Our objective is now to understand what generates such a pressure difference across a semipermeable membrane separating solutions with different osmolarities. To begin our Figure 1. Classical demonstration of osmosis. (a) A U-tube with a semipermeable membrane separating pure water on the left from an aqueous solution with an impermeant solute of concentration c s on the right. (b) With time, water will move from left to right, elevating the column of solution on the right, until its gravitational weight stops the flow. (c) Alternatively, the flow of water can be prevented if a piston applies a pressure equal to RTc s (in the dilute regime). analysis, we review first the hydraulic flow of water in response to a hydrostatic pressure difference and relate this motion to that induced by a difference in osmolarity. We consider a membrane with pure water on both sides when a transmembrane hydrostatic pressure difference ΔP is imposed (for example with a piston). The volume of water flux per unit area of a membrane is given by the empirical relationship (Weiss, 1996).
with the water flux being directed to the side with lower pressure and L p is the hydraulic permeability. The value of L p depends on the specific composition and structure of the membrane that allows water to move across it. Eq. 2 is Darcy's law, which can be derived from the Navier-Stokes equation for the convective flow of a liquid. 2 The volume water flux across a semipermeable membrane subject to both a hydrostatic pressure difference and a difference in osmolarity can be derived by combining Eqs. 1 and 2, Eq. 3 has a long history and has been proposed by many scientists in different fields, sometimes only in words. It is sometimes called Starling's equation in physiology (Starling, 1896;Blaustein et al., 2019), and for dilute solutions with impermeable solutes, it is part of the Kedem and Katchalsky (1958) equations, but it could without exaggeration be called the "Fundamental Law of Osmosis." A remarkable feature of Eq. 3 is that two physically distinct driving forces, an imposed hydrostatic pressure difference ΔP and an osmolarity difference RTΔc s , produce the same flux of water. The connection between force and flow is given by the same coefficient L p in both cases. The implication for the underlying physical mechanisms of pressure and osmotic flow is that these mechanisms must be one and the same.
Note also that van 't Hoff's law at equilibrium is recovered from the Fundamental Law by setting the flux Φ V equal to zero. If the coefficients for the two driving forces were different, van 't Hoff's equation would be violated.
When the volume flux is carried only by the water, the number of moles of water flowing across a unit area of membrane can be derived for dilute solutions from the molar volume of water v 0 w (Finkelstein, 1987): Substituting Eq. 3 into Eq. 4 gives an alternative form of the Fundamental Law of Osmosis, where P f is the osmotic permeability coefficient, P f can be determined from the measurement of water fluxes induced either by a hydrostatic pressure difference or a difference in osmolarity across a membrane (Fettiplace and Haydon, 1980;Finkelstein, 1987). The foregoing observations give rise to several questions, which we will pick up later. What is the physical reason for the observed equivalence of hydraulic and osmotic flow? It is counterintuitive that the same coefficient, L p or P f , should apply to both. Why, from a molecular point of view, must an impermeable solute concentration be balanced at equilibrium by a difference in hydrostatic pressure, and why should van 't Hoff's law be so similar to the equation of state of an ideal gas?
How water moves across membranes The flow of water is composed of two components, a convective component and a diffusive component (Truskey et al., 2009). Both may be present simultaneously but to different degrees depending on the nature of the flow. For macroscopic flow, the convective movement dominates, but we will give an example of flow through a lipid bilayer that is entirely diffusive. We will describe the convective and diffusive contributions in turn.
Convection is the bulk flow of liquid induced by a force. It is what we are able to see when water runs in a brook or through a pipe and is described mathematically by the Navier-Stokes equation (Truskey et al., 2009;Phillips et al., 2012). At the molecular level, in convective flow, clusters of closely packed water molecules move in concert in the direction of the force. However, because molecules in a liquid can move relative to each other, they are always in random motion, which drives diffusive movement. If, in addition to thermal motion, a mechanical force F acts on the molecules, their random movements are biased in the direction of the force, and each molecule acquires a drift velocity μF in the direction of the force. The proportionality constant μ is called the diffusional mobility of the molecule, 3 and it is connected to the diffusion constant D through the Einstein relation μ = D/RT. Molecules within a liquid flowing convectively under a force therefore simultaneously exhibit diffusive motion that is superimposed upon the convective flow. More specifically, the average velocity of a molecule in a flowing liquid is the sum of the convective velocity and the diffusive drift velocity.
As an example, a pressure gradient in water simultaneously induces both convective flow according to the Navier-Stokes equation and a diffusive drift of water molecules along the gradient. Clusters of water molecules move as a whole along the pressure gradient, while each individual molecule responds to the gradient by drifting stochastically away from regions of higher pressure and toward regions of lower pressure. The reason that an individual molecule drifts toward a region of lower pressure is that less work is required at lower pressure to accommodate the molecular volume.
For flow through membranes, we can quantify the relative importance of the convective and diffusive contributions with the dimensionless P f /P d ratio. The overall permeability P f has already been defined as characterizing the flow observed when a pressure or osmolarity difference is imposed on the two sides of the membrane in accordance with the Fundamental Law of Osmosis, Eq. 5. The diffusional permeability coefficient P d is what the permeability would be in the absence of convection. Then, only the diffusion of the water molecules is effective in the transport. Significantly, P d can actually be measured in a separate experiment from the observed diffusional flux ϕ * w of trace concentrations of isotopically labeled water (Mauro, 1957;Fettiplace and Haydon, 1980), in the absence of either a pressure or osmolarity difference, and Δc * w is the difference in concentration of the water isotope across the membrane. 4 That the P d in this equation is the same P d appearing in the P f /P d ratio requires proof, which is provided in Appendix 2.
It is likely that the diffusional and convective flows of water are additive, so we write P f P c + P d , where P c is the contribution from convection, and then when we divide both sides by P d , we find that for the P f /P d ratio, from which a useful interpretation of P f /P d emerges. From its meaning, convection is represented by a positive value of P c , the smallest possible value of P f /P d is unity, and then the flow is entirely determined by the diffusion of the water molecules.
But if P f /P d is much greater than unity, convection dominates osmotic flow through the membrane. Mauro (1957) and Robbins and Mauro (1960) measured P f and P d for a series of synthetic collodion membranes of increasing density in polymer material. For the most open membrane, the diffusive component of water flow was a small fraction, 1/730 of the overall observed flow, while for the mostdense membrane, the diffusive contribution was somewhat more important, but still just 1/36 of the total. Their experiments showed conclusively that the water flow in these membranes is dominated by convection, like water running in a brook, perhaps obstructed in its course by rocks (in the membrane, by polymer material).
Unlike most synthetic membranes, biological membranes are heterogeneous, with protein channels like aquaporin spanning the lipid bilayer (White et al., 2022). Water is transported independently through both the bilayer and the channels, as illustrated in Fig. 2. The P f /P d ratio provides insight in the biological case also. For isolated lipid bilayers, measurements show P f /P d 1 (Fettiplace and Haydon, 1980), so there is no convective flow component. Water crosses the lipid bilayer diffusively as dispersed independent molecules. However, the measurements of Hevesy et al. (1935) in frog skin many years ago showed that P f was greater than P d . This inequality was also found to be true in red blood cells (Paganelli and Solomon, 1957). These experiments provided the first evidence of water channels; however, it took a long time to identify and isolate aquaporin channels (Agre et al., 1995).
There is no convective (Navier-Stokes) water flow in the strict sense through aquaporin channels since the water molecules move in a single file. Nonetheless, the molecules are thought to be in close proximity in the channel, and observed values greater than unity of the P f /P d ratio could reflect their influence on each other during osmotic flow.

Common misconceptions about osmosis
Diffusion is not the primary driver of osmosis. A major obstacle bedeviling our understanding of the molecular level of osmotic pressure and osmosis, for well over a century, is the belief that diffusion is the sole driver of osmosis. Here is a typical statement: "Water spontaneously moves 'downhill' across a semipermeable membrane from a solution of lower solute concentration (relatively high water concentration) to one of higher solute concentration (relatively low water concentration), a process termed osmosis or osmotic flow. In effect, osmosis is equivalent to 'diffusion' of water across a semipermeable membrane" (Lodish et al., 2021). Or, "…water moves slowly into or out of cells down its concentration gradient, a process called osmosis" (Alberts et al., 2015).
Although water diffusion may seem to provide a reasonable mechanism for osmosis, measurements from membranes with aqueous water channels show conclusively that diffusion alone cannot account for the osmotic flux. The fact that P f > P d demonstrates that there is a significant convective component to osmotic water fluxes that cannot arise by diffusion. This disparity points to the need for a driver in addition to the water gradient. This is precisely what the Debye model does, showing how the collision of the solute molecules with the membrane generates a pressure drop that drives water across the membrane.
It is incorrect to characterize the osmotic flow of water as essentially a Fick's law diffusion of water molecules between aqueous solutions of differing water concentrations. The difference in water concentration (moles per unit volume) in pure water and in an aqueous solution is not simply a function of solute concentration alone. A straightforward calculation shows that it also depends on the ratio of the partial molar volume of the solute species to the molar volume of pure water (see Appendix 3). This ratio is specific to the particular solute species. The same concentration of solute, but for different solute species, leads to water concentration differences between the two solutions that are specific to the specific solute species. If the osmotic flow were caused by the difference in water concentrations, the water flux would then be specific to the solute species used to establish it. Such a dependence on impermeable solute species is not observed for dilute solutions, and moreover, would contradict both van 't Hoff's equation and the Fundamental Law of Osmosis.
The mechanism of osmosis cannot be inferred from the properties of free-standing solutions. Another misconception arises from a focus on the bulk properties of the solutions bathing the membrane, while ignoring the physical implications of the most obvious property of the membrane itself, namely, its mechanical interaction with the solute making it impermeable to the solute molecules. The most common mistake, which has recurred persistently, is the idea that in a free-standing solution, the solute and solvent possess independent pressures, just like a mixture of ideal gases. Modern thermodynamic and statistical mechanical ideas of liquid solutions have fortunately taken root, and today this erroneous picture is only rarely invoked.
The modern thermodynamic analysis of osmotic pressure is correct but provides no information about the mechanism. It compares the chemical potentials of water in a free-standing solution with the chemical potential of water in pure water with no reference to the physical interaction of membrane with solute.
Osmotic transport is not different from transport induced by pressure differences. Another misconception is to deny the reality of the pressure underlying the movement of water across a semipermeable membrane. Here is an example: "The relationship (van 't Hoff) however arises directly from the parallels in the thermodynamic relationships and should not be interpreted in the molecular mechanistic sense since the osmotic pressure is in fact a property ensuring equilibrium of the solvent and solute, and has its effect only via its reduction of the chemical potential of this solvent" (Tombs and Peacocke, 1974). The identification of hydrostatic pressure-driven flow and flow driven by a concentration imbalance of impermeable solute is embodied in the Fundamental Law of Osmosis, and we will demonstrate how the Debye model explains this equivalence in a physically transparent way.
A mechanistic model for osmosis: The Debye model Several different mechanisms have been proposed to explain how osmosis arises, with Guell (1991) listing 14. 5 We will argue that there is in fact a parsimonious explanation for osmosis that relies on the mechanical interaction between the membrane and impermeable solute molecules, and that we will refer to as the Debye model as it was first proposed by Debye in 1923. Despite Debye's reputation, the model made little impact on our understanding of osmosis-disappearing for decades, probably because biologists were not aware of it and chemists and physicists were largely uninterested-until the 1960s. Unfortunately, the connection to the original work was lost and we reestablish it here (see Box 1 for a short history).
Debye recognized that the physical principles underlying the development of an osmotic pressure must be centered on the interactions of the membrane with the solute molecules since osmotic pressure is not observed in the absence of a membrane. As Debye put it in his 1923 paper, "We express the quality of semi-permeability of the membrane by saying that the potential energy of a solute molecule increases from zero to infinity when it is transported across the membrane from the solution side." An equivalent statement would be that the membrane exerts a repulsive force F on a solute molecule that is strong enough to prevent the solute molecule from entering the membrane and crossing over to the pure solvent side.

The Debye model leads to van 't Hoff's law
Debye was concerned only with osmotic equilibrium, so we begin by following his derivation of van 't Hoff's law for osmotic pressure at equilibrium. Afterward, we discuss steady-state osmotic flow as a straightforward extension of his model (Manning, 1968). Consider a semipermeable membrane separating two chambers at equilibrium, with the x coordinate increasing from left to right, the semipermeable membrane perpendicular to the x axis, the solution compartment with solute concentration c s,r to the right of the membrane, and the pure solvent to the left (Fig. 3 A). We are in effect looking at an infinite 2-D membrane, with all values isotropic in the y and z directions.
Our first goal is to obtain an equation characterizing the solute concentration profile c s (x). For that, we write an equation for the flux j s of solute molecules in the absence of applied pressure, where the first term on the right-hand side of the equation is Fick's law for solute flux in the presence of a solute concentration gradient in dilute conditions, and the second term is the contribution to the solute flux from the mechanical force F exerted by the membrane on nearby solute molecules. Einstein's relation D = RTμ (Einstein, 1905) will allow us to convert the solute mobility μ to its diffusion coefficient D. The semipermeability property of the membrane means that passage of solute into and through the membrane is completely blocked by the force F . Therefore, there must be a gradient of solute concentration across the membrane-solution interface where from left to right the solute concentration increases steeply from zero just inside the membrane to the constant value c s,r of solute concentration in the solution chamber. Moreover, since the membrane excludes the solute, the solute flux across the interface must vanish. Setting j s = 0 and then using Einstein's relation and canceling D, we obtain an equation to characterize the solute concentration profile c s (x), To connect this equation to the pressure that develops across the membrane, we can visualize a volume element of the solution near the membrane as a thin slice of thickness dx parallel to the membrane (see rectangular blue box in Fig. 3). When the system is at equilibrium, the slice, in particular, must be in mechanical equilibrium, meaning that all of the forces acting on and inside the slice must balance out to zero. The intermolecular forces among the molecules inside the slice cancel each other as a consequence of Newton's law of action-reaction, leaving the requirement that the forces on the slice originating from outside it must balance it to zero. These forces are the repulsive force F from the membrane acting on each solute molecule in the slice and the hydrostatic pressures from the fluid surrounding the slice and pushing from outside the slice on each of the side surfaces of the slice. With P(x), the pressure at x, the zero balance is expressed by the equation dP/dx − c s F 0 or 7 dP dx c s F .
Eq. 10 and Eq. 11 can be combined: The van 't Hoff equation for osmotic equilibrium, follows after integration from left to right (pure solvent to solution) with P r the pressure in the solution compartment, P 0 the pressure in the pure solvent compartment, and of course c s,l = 0 in the pure solvent compartment.
We are now in a position to recognize the genius of Debye's insight, simple as it is. At the heart of his derivation is the membrane-solute force F , which would be different for every membrane and every solute. How can it lead to van 't Hoff's equation, which is applicable generally to any membrane-solute pair? The reason, as we have just seen, is that it produces compensating physical effects, and F cancels from the final result.
It is worthwhile considering an alternative approach, first used by Ehrenfest (1915) and then by others (Kiil, 1982;Borg, 2003 Preprint;Bowler, 2017) to employ the virial theorem to understand osmosis. The virial theorem from the statistical mechanics of a fluid is a relation between the pressure of the fluid and its total time-averaged energy, kinetic plus potential. The potential energy accounts for the forces of interaction among the particles of the fluid. For a real gas, the virial theorem was developed by Mayer (Uhlenbeck and Ford, 1963) into his virial expansion, an infinite series for the pressure in which the first term gives the ideal gas equation of state and the higherorder terms account successively for corrections due to interactions among the gas molecules. The McMillan and Mayer (1945) theory gives an analogous virial expansion for the osmotic pressure that arises when a solution is separated from Box 1. A short history of the Debye model The investigation of osmosis has an interesting history that has been told by others (Smith and Smith, 1960;Hammel and Scholander, 1976;Mason, 1991). In this section, we will focus on the history of the Debye model.
Although the experimental demonstration of osmosis by Jean-Antoine Nollet (1748) predates that of diffusion by Thomas Graham (1828), the development of the theoretical basis of diffusion proceeded with little controversy (Einstein, 1905;Jacobs, 1935;Berg, 1993). In contrast, the theoretical underpinnings of osmotic pressure proved contentious from the start. 6 There is a fascinating story recounted by Wald (1982) that it was Hugo de Vries (a botanist and one of the rediscoverers of Gregor Mendel's work) who told van 't Hoff about Pfeffer's experiments (Pfeffer, 1890) on semipermeable membranes when their paths crossed while walking in Amsterdam. van 't Hoff was awarded the first Nobel Prize in Chemistry in 1901 largely for his work on osmosis. At our historical remove, it may seem strange to award the prize for what seems like such a simple finding. However, it provided one of the first experimental confirmations of atomic theory. What we have called the Debye model was first proposed by Peter J.W. Debye in a paper first published in French (Debye, 1923b) and then in German (Debye, 1923a), and primarily devoted to further developments of Debye's theory of ionic solutions. Debye remarks in a footnote "Among the large number of authors who have already dealt with the kinetic theories of osmotic pressure, we must cite above all: L. Boltzmann, H.A. Lorentz, Ph. Kohlstamm, C. Jäger, O. Stern, P. Langevin, J.J. van Laar, P. Ehrenfest," but does not cite any of their papers, because they failed to pin down the mechanism.
In the intervening years, there have been very few references to Debye's paper. Joos developed a simplified derivation of the mechanism in what is essentially a didactic paper (Joos, 1941), acknowledging that his work was derived from an idea in a paper by Debye (1923b). The derivations were included in Joos's influential textbook of physics (Joos and Freeman, 1959). Manning (1968) was probably the first to rederive the Debye model in the second half of the 20th century. Manning based his derivation on a textbook by Rutgers (1954), who said that his argument was derived from Debye, but Rutgers does not quote the paper. It is worth noting that Debye provided a foreword to the Rutgers textbook.
The textbook by Villars and Benedek, 1974 is the source most often quoted for the solute-membrane repulsion model, but it has no references at all. In the biological literature, Mauro (1979) appears to be the first to have referred to Manning and to Villars and Benedek in the context of osmosis.
It is puzzling that Debye's work on osmosis made little impact since he was a major figure in the development of physics in the 20th century, receiving the Nobel Prize in 1936. It is even more so because he was a professor at Cornell University (Ithaca, NY, 1939-66) during the period when the debate about the molecular origins of osmosis was revived. Indeed, from the mid-1950s to the 1990s, several theories competed about the origin of osmotic pressure (Hammel, 1979;Hildebrand, 1979;Mauro, 1979;Soodak and Iberall, 1979;Yates, 1979;Essig and Caplan, 1989). Prominent among the contesting theories was the controversial solvent tension theory (Hammel and Scholander, 1976). However, the Debye model never seemed to have made an appearance in the debate, at least in its quantitative form. In an interview in 1964, Debye himself provides a possible key to this enigma. When asked which periods of his work stand out to him "…I think they are important at the moment when I am doing them. Later I forget about them. So it's only during the time that I have fun with them that they seem important" (Corson et al., 1964).
6 "Again we have the basically pointless question: What exerts osmotic pressure? Really, as already emphasized, I am concerned in the end only with its magnitude; since it has proved to be equal to the gas pressure one tends to think that it comes about by a similar mechanism as with gases. Let he, however, who is led down the false path by this rather quit worrying about the mechanism."-van 't Hoff (1892) translation from Weiss (1996). pure solvent by a semipermeable membrane. The first term of the expansion gives the van 't Hoff equation and the higherorder terms account successively for solute-solute interactions as mediated by the solvent. However, this approach cannot provide any insight into the membrane-solution forces that generate pressure. It is effectively equivalent to the thermodynamic analysis of osmosis using the chemical potential of water (see above).

The Vegard pressure profile
We now move from considering osmotic equilibrium to the situation where the pressures in the chambers are constrained to be the same and both chambers are very large and well stirred. Under these conditions, which we will refer to as the osmotic steady state, an osmolarity gradient across the semipermeable membrane will drive a steady flow of water across the membrane. We will show that extension of the Debye model to osmosis demonstrates that there must be a pressure drop from the solution to just inside the membrane equal to RTc s . Since the pressure is lower on the solution side (just inside the membrane) than on the pure solvent side, there is necessarily a pressure gradient across the membrane. In a simple 1-D visualization, the expected pressure profile is shown in Fig. 3 a. However, the pressure gradient within the membrane may have a more complicated form shaped by the molecular structure of the membrane.
In a prescient 1908 paper, Lars Vegard, who was a student of J.J. Thompson at the time, appears to have been the first to propose this pressure profile (Vegard, 1908). He suggested, based on osmotic transport measurements with synthetic membranes, that somehow the solute generated a pressure gradient within the membrane but did not propose a mechanism. Such pressure profiles were rediscovered by several workers (Dainty, 1965;Mauro, 1965;Manning, 1968) in the 1960s. Manning first made the connection between the profile and the Debye model (see Fig. 3 C of Manning [1968]). We term this peculiar pressure profile the Vegard pressure profile and the pressure drop in the narrow interface region on the solution side the Vegard pressure drop.
The Vegard pressure profile provides a graphic description of the force that drives the osmotic flow of water. The intramembrane pressure gradient drives water from the side with the lower osmolarity (pure solvent in Fig. 3) to the side with the higher osmolarity. In the narrow interface region on the high osmolarity side, the pressure drop by itself would drive water back toward the membrane, but in this region the Debye model shows that it is balanced by the forces from the membrane that drive the impermeable solute molecules away.

The Vegard pressure drop drives osmosis
With reference to Eq. 11 and the discussion above it, we have explained that the difference dP/dx − c s F represents the net force on a volume element of the solution near the membranesolution interface and at equilibrium it equals zero. When the system is not in equilibrium, the difference dP dx − c s F dV is still the total net force on a volume element dV at the membranesolution interface, but it is not zero and gives rise to a volume flux Φ V . If the flux is not too large, we can set down a linear relationship between the net force and the volume flux, where we will verify the identification of the proportionality constant as hL p , where h is the width of the membrane and L p is the permeability in Darcy's law, Eq. 2. The relation between the membrane force F and the solute concentration gradient at the membrane-solution interface, Eq. 10, remains valid in the steady-state case since we assume well-stirred conditions at the interface so that this expression for Φ V becomes Now, we integrate both sides of this equation across the membrane-solution interface from just inside to just outside. The integral involving the volume flux Φ V is small because it is proportional to the narrow width of the interface. But the integrals of the pressure and concentration derivatives do not depend on the width of the interface. The integral of the pressure derivative across the interface equals the pressure difference across the interface. The integral of the derivative of solute concentration equals the difference of solute concentrations across the interface. This latter difference equals the bulk solute concentration in the solution because the concentration of impermeable solute just inside the membrane is zero. The result then of integrating both sides of Eq. 15 across the membrane-solution interface is that from outside to inside there is a pressure drop equal to RTc s,r at the interface. In other words, the pressure just inside the membrane on the solution side is lower by this amount than the pressure P 0 of the solution outside. Since the pressure is P 0 in both chambers outside the membrane, there must be a pressure gradient across the entire membrane from P 0 on the pure solvent side to P 0 −RTc s,r on the solution side, and hence we have produced the Vegard pressure profile and pressure drop.
We can take the derivation one step further, and in doing so, both illuminate the action of the pressure gradient and verify the choice of coefficient hL p . The solute concentration is zero inside the membrane and so its gradient is also zero there. Setting dcs dx 0 in Eq. 15, we see that inside the membrane, an equation that explicitly exhibits the volume flux as driven by a pressure gradient inside the membrane when the pressures in both solution and pure solvent compartments are equal. Moreover, with these coefficients, this equation is equivalent to Darcy's law (Eq. 2).
Applications of the Debye model to osmotic flow through biological membranes Stiff porous membranes The Debye model, based as it is on fundamental physical principles, should be applicable to osmotic flow across any pressure-bearing membrane, including synthetic polymer-based membranes, the copper-ferrocyanide membrane used by Vegard, and the collodion membranes in Mauro's measurements. In the latter, a P f /P d ratio much greater than unity suggests a pressure-driven bulk water flow inside the membrane with Debye-Vegard pressure drops at the solution-membrane interfaces and a pressure gradient traversing the membrane of the Vegard type (see Fig. 3). These synthetic membranes should be realistic models for biological structures such as the walls of microvessels. The smallest pores crossing capillary walls are about 5 nm wide (Michel and Curry, 1999), much larger than a water molecule (∼0.3 nm), thus carrying water in more or less its ordinary bulk liquid form. The osmotic water flow across capillary walls is hence expected to be consistent with the Debye-Vegard model.

Cell membranes
Plant cell membranes are supported by a pressure-bearing cell wall and the Debye model for osmotic flow is expected to hold true. Although animal cell membranes lack a cell wall, they are reinforced by a submembrane cytoskeletal network (Kapus and Janmey, 2013).
The lipid bilayer in some biological cell membranes is spanned by aquaporin water channels (Preston et al, 1992;Walz et al, 1997). Lipid bilayers are very permeable to water (Fettiplace and Haydon, 1980); however, in particular cells, the water flux is accelerated by specific aquaporins, but not all cells express aquaporins (Verkman, 2012). Since proteins are relatively stiff (Krieg et al., 2018), the Debye model is expected to account for the osmotic flow through aquaporins, just as it does through any pressure-bearing semipermeable membrane. There are Debye-Vegard pressure drops at both ends of the channel, with the larger drop occurring at the end abutting the solution of greater osmolarity. The two ends of the channel could face unequal pressures and the water molecules in the interior of the channel are therefore subjected to a force directed toward the lesser of the two pressures.
Aquaporin channels are very narrow with cross-sectional areas just sufficient to accommodate a single water molecule. The single-file movement of a column of water through such channels cannot be described as bulk convective flow, even though experimental measurements show that the P f /P d ratio is significantly greater than unity. Although the osmotic movement of water across these channels may be pressure-driven, a precise description of the dynamics of the water molecules inside the channel is a subject of current investigation (Kavokine et al., 2020).

Lipid bilayers
Lipid bilayers self-assemble in vitro and may be studied in isolation. Their P f /P d ratios are found to be equal to unity, indicating that water inside them exists, and flows, as independent molecules. Although the parallel arrangement of hydrocarbon tails permits their diffusion within the plane of the bilayer, facilitating the passage of water, out-of-plane movements of the tails are more constrained and therefore may be compatible with an internal pressure gradient. Since the external pressures on the two sides of a bilayer membrane may be equal when water transport occurs in response to a difference in osmolarity, we think the possibility of the Debye-Vegard pressure drop and interior pressure gradient are realistic.
We show in Appendix 4 that the Debye model for a lipid bilayer leads to the result P f KD/h, where K is the partition coefficient (the ratio of water concentration inside the membrane to that outside), D is the self-diffusion constant of independent water molecules inside the membrane, and h is the thickness of the membrane. Since it is clear from inspection that the same result holds for P d , we conclude that P f /P d 1, in agreement with experimental measurements.

Discussion
Our primary objective in this paper is to provide a persuasive argument for the Debye model grounded in well-established principles of physics. It begins with the Fundamental Law of Osmosis which implies that whatever happens to drive water across a membrane in the presence of an osmotic gradient must be the same as for the pressure-driven flow in the absence of an osmotic gradient. The Vegard pressure drop, on the side of the membrane adjacent to the solution with the higher osmolarity, provides a plausible mechanical basis for the law, since the osmotic flow is then also pressure driven.
A number of scientists have given verbal accounts that accord well with the Debye mechanism and are worth recalling: "To the extent that it is possible to visualize molecular events, this process could perhaps be pictured (at least for narrow pores) as a molecular piston pump, with solute molecules playing the role of the piston" (Dainty and Ferrier, 1989). And from the great epithelial physiologist Hans Ussing: "The pore contains pure water all the way through, so the driving force cannot be a difference in the chemical activity. Obviously, the driving force is the 'suction' created by the osmotic pressure difference at the dotted line. But suction is only another word for hydrostatic pressure difference" (Ussing and Andersen, 1955).
Physiologists often refer to what is termed the "colloid osmotic (or oncotic) pressure," which is the osmotic pressure that can be attributed to blood plasma proteins (Boron and Boulpaep, 2016). As blood flows into a capillary bed, the hydrostatic pressure filters plasma into the interstitial fluid leaving behind the impermeant proteins in the blood. This has the effect of decreasing the osmolarity of the interstitial fluid relative to the blood. As blood flows out of the capillaries, the hydrostatic pressure declines and now the osmotic gradient across the capillary wall drives interstitial fluid back into the blood. This interaction between hydrostatic and osmotic gradients, which is of immense importance in clearing the interstitial space, was first postulated by Starling. Although the term "colloid osmotic pressure" is useful in physiology, its mechanistic origins can also be accounted for by the Debye model. However, it is worth noting that a protein molecule contributes more than a small solute molecule to the osmolarity through an excluded volume effect (Guttman and Anderson, 1995).
Our focus has primarily been on the physical basis of osmosis, but there are several allied phenomena and concepts that we have not touched on which are worth mentioning for readers interested in exploring further ramifications of osmosis, namely: depletion forces (Asakura and Oosawa, 1958), diffusioosmosis (Marbach and Bocquet, 2019), osmotic stress (Parsegian, 2002), reflection coefficients (Finkelstein, 1987), and virial coefficients (Neal et al., 1998).
The history of attempts to find a molecular basis for osmosis is surprisingly long and tangled for what on the surface seems like a simple phenomenon. One of the primary difficulties with establishing the physical basis of osmosis is setting up the initial scenario and isolating the essential forces at play. The picture that emerged from the Debye model raised hackles and unfounded thermodynamic arguments were used to counter it. What made this situation even more complicated is that there appeared to be no way of testing the predictions of the theories. After a flurry of activity with no resolution, the debate died out, leaving the erroneous water concentration gradient model uncontested in some textbooks. An odd element that added to the confusion is that even wrong arguments led to the van 't Hoff equation.
It is worthwhile comparing the evolution of our understanding of diffusion to that of osmosis. In the case of diffusion, Einstein's explanation in 1905 was rapidly confirmed by Jean Perrin's experiments in 1909(Perrin, 1909, 1910. In contrast, it has taken a very long time for a consistent mechanistic account of osmosis to emerge. To add to that the absence of experiments addressing the osmotic mechanisms at the nanometer scale has perhaps retarded the acceptance of the Debye model. Molecular dynamics provides a method for exploring what occurs at a molecular level in a phenomenon like osmosis (Roux, 2021). In molecular dynamics, which is now a well-established discipline in molecular physics, Newton's laws of motion are used to computationally model the collisions of individual molecules. Molecular dynamic simulations using simple particles to represent the solvent and solute together with an energy barrier to model the membrane successfully recapture van 't Hoff's law (Murad and Powles, 1993;Zhu et al., 2002;Luo and Roux, 2010;Lion and Allen, 2012). This confirms that the nature of the solvent and solute are irrelevant in generating an osmotic flux. However, molecular dynamics has not been used to model the Vegard pressure profile in steady-state osmosis but has been used to predict P f /P d from the molecular structure of aquaporins (Zhu et al., 2002;Portella and De Groot, 2009) and to visualize the pressure drop within a polyamide membrane when hydrostatic pressure is applied across it (Wang et al., 2023) With the development of techniques that allow one to probe below the nanometer scale, the precise molecular mechanics of osmotic transport and the validity of the Debye model should be within reach of experiments. It is not inconceivable that molecular sensors could be designed to detect the pressure gradient's presence and extent. It should therefore be possible to probe the pressure profile first postulated by Vegard in 1908, to confirm a simple and unified view of the physical basis of osmosis.

Appendix 1
The Donnan effect Since the Donnan effect plays an important role in water transport in cells, it is worthwhile delving into its nature. To do this, we consider a simplified model introduced by Post and Jolly (1957). Let us consider a spherical "cell" with a pliant membrane that is permeable to an uncharged molecule A and water. If we place the cell in an infinite bath with A at a concentration of [A] e , and assume that the passage of A into and out of the cell is governed by the same first-order rate constant k, then: Therefore, at equilibrium cell will be stable. Now, if we introduce b moles of an uncharged impermeant molecule B into the cell, Eq. 17 remains unchanged, but the equation for osmotic equilibrium becomes: where w is the volume of the cell. The cell must follow the osmotic constraint and the kinetic constraint, and the only way that it can do this is if w→∞. So, water flows in continuously and the cell volume grows without ceasing. Although we have shown the case for an uncharged molecule, the same holds true for charged solutes. The volume can be stabilized by introducing an impermeant molecule in the extracellular space. However, this is not what animal cells do; instead, they pump molecules out of the cell to stabilize cellular volume (Tosteson and Hoffman, 1960). In the case of the toy model that we have introduced here, it can be shown that if A is pumped out of the cell, the volume can be stabilized in the presence of B.

Appendix 2
Identification of P d Eq. 7 is a statement of Fick's law for the diffusion of tracer molecules when there is a concentration gradient of the tracer. Therefore, P d D/h, where D is the self-diffusion constant of water molecules in the membrane and h is the membrane thickness. The question is whether this P d also characterizes the diffusive component ϕ w,d of water flux, not tracer molecules, when a force per molecule f is imposed. The answer is yes, as we show here. Note that an arbitrary multiplicative factor, such as a partition coefficient, does not affect the conclusion. Using the Einstein relation between diffusion constant and diffusional mobility, we have ϕ w,d = (D/RT) (N w /Ah)f, where N w is the number of water molecules in the membrane and Ah is the volume occupied by the membrane, A being the cross-section area and h the membrane thickness. The total force F on the water in the membrane is N w f, and D/h P d . Then, ϕ w,d P d (F/A)/RT), where we use a molar flux. The net force per unit area can be imposed by a pressure difference and then ϕ w,d −P d ΔP/RT, completing the proof.

Appendix 3 Relation between water and solute concentrations
The sum of the water and solute concentrations c w + c s is (n w + n s )/ V, where n i is the number of moles of species i and V is the volume of solution, equal to (n w v w + n s v s ), where v i is the partial molar volume of species i. A straightforward rearrangement leads to where X s [=n s /(n w + n s )] is the mole fraction of solute. Only if the solute species is essentially identical to water, for example, D 2 O, can we say v s v w v 0 w , where the latter is the molar volume of pure water, and thus obtain from this equation the simple result c w + c s = 55.5 M. In this situation, the water concentration depends only on the solute concentration and is independent of the specific solute species. In general, however, the concentration of water and that of solute are not simply related.